A355635 Triangle read by rows. Row n gives the coefficients of Product_{k=0..n-1} (x - binomial(n-1,k)) expanded in decreasing powers of x, with row 0 = {1}.

1, 1, -1, 1, -2, 1, 1, -4, 5, -2, 1, -8, 22, -24, 9, 1, -16, 93, -238, 256, -96, 1, -32, 386, -2180, 5825, -6500, 2500, 1, -64, 1586, -19184, 117561, -345600, 407700, -162000, 1, -128, 6476, -164864, 2229206, -15585920, 51583084, -64538880, 26471025

Offset

0,5

Comments

Without signs the triangle of elementary symmetric functions of the terms binomial(n,j), j=0..n.

Links

Table of n, a(n) for n=0..44.

Formula

T(n, 0) = 1.

T(n, 1) = -2^(n-1), for n > 0.

T(n, 2) = A000346(n-2), for n > 1.

T(n, 3) = -A025131(n-1), for n > 1.

T(n, 4) = A025133(n-1), for n > 1.

T(n, n) = (-1)^n*A001142(n-1), for n > 0.

T(n+1, n) = (-1)^n*A025134(n).

T(n+2, n) = (-1)^n*A025135(n).

Example

The triangle begins:
  1;
  1,  -1;
  1,  -2,   1;
  1,  -4,   5,    -2;
  1,  -8,  22,   -24,    9;
  1, -16,  93,  -238,  256,   -96;
  1, -32, 386, -2180, 5825, -6500, 2500;
  ...
Row 4: x^4 - 8*x^3 + 22*x^2 - 24*x + 9 = (x-1)*(x-4)*(x-6)*(x-4)*(x-1).

Prog

(PARI) T(n, k) = polcoeff(prod(m=0, n, (x-binomial(n-1, m))), n-k+1);

Crossrefs

Cf. A000079, A000346, A025131, A025133, A025134, A025135.

Cf. A001142 (right diagonal unsigned).

Sequence in context: A158471 A158472 A198895 * A118686 A355540 A102610

Adjacent sequences: A355632 A355633 A355634 * A355636 A355637 A355638

Keyword

sign,tabl

Author

Thomas Scheuerle, Jul 11 2022

Status

approved